MATHEMATICAL MODELS ON THE GROWTH OF SOLID TUMORS - Quantitative Medical Data Analysis Using Mathematical Tools and Statistical Techniques - page 310

easier. Then one provides the theoretical framework for modeling at cellular
scale in such way that the corresponding models at extracellular scale and
cellular prediction should be close. If the cellular scale model is designed
properly, one may hope it covers not only the extracellular behavior of the
system in question, but also some of its cellular scale features. The cellular
model by its nature is richer and it describes a larger variety of phenom-
ena. In a similar manner the subcellular scale model should be richer than
cellular and extracellular scale models.
This survey refers to a general framework for a program for nding
transitions between the dierent scales of descriptions, interacting entities
(cellular, subcellular), statistical description of test entity, and macroscopic
scale (extracellular).
In mathematical terms the links of the following mathematical struc-
tures was developed for various situations of biological interest 43;44;45;46;47 :
(1) The micro-scale of stochastically interacting entities (cells, individu-
als,..), in terms of continuous linear semigroups of Markov operators
(continuous stochastic semigroups) 48 ;
(2) The meso-scale of statistical entities in terms of continuous nonlinear
semigroups related to the solutions of bilinear Boltzamann-type nonlo-
cal kinetic equations 49 ;
(3) The macroscopic scale of densities of interacting entities (in terms of dy-
namical systems related to bilinear reaction-diusion-chemotaxis equa-
tions.
Lachowicz 46 deals with the mathematical theory of a large class of
reaction-diusion systems (with small diusion) and then generalized to
include reaction-diusion-chemotaxis systems. This was motivated by a par-
ticular model of tissue invasion by solid tumor reaction-diusion equations
with a chemotaxis-type term 30;55 . The model is quite general and can be
applied to a large class of systems at the macroscopic level including the
Keller-Segel-type systems.
There is a huge literature related to the rigorous derivation of chemo-
taxis equations from cellular scale models 54 . Stevens 66 proved that for suf-
ciently large numbers of particles the dynamics of an interacting particles
system can be approximated by the solutions of chemotaxis systems.
Later, Lachowicz proposed a more general approach in the sense that
it can be applied to large class of models at the macroscopic scale. More-
over it relates the three scales of descriptions. The methods may lead to
new and more accurate modeling of complex process, like tumor growth.